All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
1
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Quick question about conjugate equivalence bimodules and inner products
Let $A$ and $B$ be $W^{*}$-algebras, let $X$ be an $A-B$-equivalence bimodule (according to the definition given in "Morita equivalence for $C^*$-algebras and $W^*$-algebras" by Rieffel, link:http://...
- 417
3
votes
1
answer
248
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example of a compact quantum group at a root of unity?
In Woronowicz's theory of compact quantum groups, the most well-known example is $SU_q(2)$, for $q$ a real number. Moreover, all the other examples of compact quantum groups, based some Drinfeld--...
- 31
1
vote
0
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137
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A continuous choice of invertible elements
Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology.
Is there a continuous map $\alpha:A^...
- 356
9
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1
answer
973
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"Averaging" in von Neumann algebras
I've been reading recently about various "averaging" tricks in von Neumann algebras. For example, Christensen and Sinclair, in "On von Neumann algebras which are complemented subspaces of $B(H)$." J. ...
- 18.7k
2
votes
1
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443
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crossed product
on Williams Crossed product book,on page 198, it is mentioned that there is only one regular representation for C_c(G), and that is the left regular representation.
I know that this representation is ...
- 23
7
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2
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330
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Perturbation in C*-Algebra
Let x be an element in a C*-algebra A, is it true that if x approximately commute with every element in A, then x is near the centre of A? More precisely, I want to know whether the following is true： ...
- 411
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0
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481
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Versions of the spectral theorem
Since any $C^*$-algebra can be represented as an algebra of bounded operators $\mathcal{B(H)}$ on a Hilbert space $\mathcal{H}$, the spectral theorem applies to all $C^*$-algebras:
($*$) $A=\int_{\...
- 585
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1
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287
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All AI-algebras are AT-algebras
It is known that every AI-algebra (i.e. inductive limit of interval algebras) is an AT-algebra (i.e. inductive limit of circle algebras)?
This seems a little bit odd because a building block of an AT-...
- 169
5
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2
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408
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Derivation of von Neumann algebra which is zero on MASA
Are there any example of $II_1$-factor $M$ with maximal abelian von Neumann subalgebra $A$ and non-zero derivation $\delta:M\rightarrow B(H)$ such that $\delta(a)=0$ for every $a\in A$?
- 4,705
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1
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323
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Ideal spanned by matrix units isomorphic to compact operators
Hello,
Assume we have $(n+1)$ isometries $S_1,...,S_{n+1}$ in the separable Hilbert space $H$ with the properties that $\sum_{i=1}^{n+1}S_iS_i^*=I, S_i^*S_j=0$ (i.e. $S_i$ are the generators of the ...
- 23
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2
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492
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trace measurable operators
Hello,
I have a question about trace measurable operators and I think it's not a hard one. However, I'm quite confused because I cannot prove it.
Let $\mathcal{M}$ be a semi-finite von Neumann ...
- 85
2
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2
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1k
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Why is this a conditional expectation into the fixed point algebra?
Let $A$ be a C*-algebra and let $\alpha$ be an action of the circle group $S_1$ on $A$ (Gauge action).
We define the following map:
$$E:A\rightarrow A;\quad E(a):=\int\alpha_t(a)\textrm{d} t.$$
My ...
- 23
7
votes
1
answer
577
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Does a crossed product R⋊_α F_n of the hyperfinite factor of type II_1 and a free group have the QWEP?
Let $\mathcal{R}$ be the hyperfinite factor of type $\rm{II}_1$ and let $\mathbb{F}_n$ be a free group with $n$ generators. Let $\alpha$ be an action of $\mathbb{F}_n$ on $\mathcal{R}$.
Does the von ...
- 1,222
13
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1
answer
1k
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Endomorphism of type III factor: can it satisfy $\phi\circ\phi = \phi\oplus\phi$?
I'm still trying to get some feeling about this question...
Given Jesse Peterson's answer to this question (he showed that $\phi\circ\phi\sim\phi$ is impossible), I suspect that the following is also ...
- 43.2k
3
votes
1
answer
194
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Local cross sections for Unitary group in a hilbert space
Let $U(\mathcal{H})$ be the group of unitary operators on a Hilbert space with the norm topology. Let $H\subset U(\mathcal{H})$ be a closed subgroup.
Under which condidions (on the ...
- 2,630
1
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1
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2k
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Square root of integral operator
Consider the 1-torus $\mathbb{T}$. Let $k$ be a smooth function on $\mathbb{T}^2$ and $K$ be the integral operator on $L^2(\mathbb{T})$ with kernel $k$. One can show that $K$ is of trace class, hence $...
- 1,702
7
votes
2
answers
441
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endomorphism of factor: can it be idempotent up to congugacy?
Let $M$ be a factor, and let $\phi:M\to M$ be an irreducible endomorphism
("irreducible" means that the relative commutant of $\phi(M)$ in $M$ is trivial).
Let's also assume that $\phi$ is not ...
- 43.2k
5
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1
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673
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Unbounded representations of Banach algebras
Can a representation of a Banach algebra be unbounded?
To clarify, I'm not asking about a representation as unbounded operators, but
rather a homomorphism $\pi: A \to B(H)$ for some Hilbert space $H$,...
- 275
3
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1
answer
1k
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Self-adjoint bounded operator, resolution of the identity, def. of the diagonal
Let $A$ be a self adjoint bounded linear operator with a continuous spectrum
$\sigma(A)=[a,b]$ which acts on a separable Hilbert space. Let
$E_\lambda$ be its resolution of the identity.
For ...
2
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1
answer
348
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Atomic enveloping von Neumann algebra
Let $A$ be a $C^*$-algebra. If the second dual of $A$, which is the enveloping von Neumann algebra of $A$, is atomic, can we deduce that $A$ is an ideal in its second dual ?
- 195
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318
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What's the link between the Toeplitz operators on H^2 and those used to define Cuntz-Pimsner algebras?
An alternate way to phrase this question might be, "How did the Toeplitz operators used in the definition of the Cuntz-Pimsner algebra come by their name?" or, "What's the relationship between the ...
- 151
3
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1
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441
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Are the categories of representations of G and C*(G) isomorphic?
Let G be a locally compact Hausdorff group, and C*(G) the full group C* algebra.
I found in some books that "representation theory of both is the same". Can this be expressed as "the categories are ...
5
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0
answers
189
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Topology for bounded operators quotiented by Schatten ideal
I saw this particular question on stackexchange. Since there has been zero answers and since I've been interested in this question myself I want to ask it here.
Given the $C^{\ast}$-algebra of bounded ...
- 1,338
2
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1
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352
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Cyclic vectors for C* algebras
Let A be a C* algebra of operators on a Hilbert space H. Can it happen that for some x in H the set Ax is dense in H but it is not the whole H?
- 51
3
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1
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385
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Idempotent homomorphisms of von Neumann algebras
Is there any description of unital idempotent ($F^2(x)=F(x)$) morphisms of a von Neumann algebra into itself? Or, equivalently, of weakly closed subalgebras which are retracts as von Neumann algebras?
- 1,152
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1
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684
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Lifting of a ucp map with values in a von Neumann algebra ultraproduct of matrix algebras
Let $u:A \to \prod_{\mathcal U} M_n$ be a unital completely positive map (ucp) from a unital separable $C^*$algebra into the von Neumann algebra ultraprodut $\prod_{\mathcal U} M_n$.
Here $\mathcal ...
- 9,486
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2
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898
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Unitary Operator as a complex valued function
A book on Quantum Mechanics states, "A unitary operator can be considered to be a complex valued function of a Hermitian operator."
Please give a hint on how to prove this assertion.
user16110
4
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0
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171
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$S^{3}$-valued harmonic analysis
Edit:
Note that $S^{3}$ with the quaternion operation is a group. For a locally compact Abelian group $\Gamma$ we consider
$$\tilde{\tilde{\Gamma}}=\{\phi:\Gamma \to S^{3} \mid \phi(xy)=\phi(...
- 356
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1
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122
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Is there an irreducible subfactor with an infinite homogeneous single chain lattice?
We know that we can build an irreducible subfactor realizing a finite single chain lattice containing any finite index irreducible maximal subfactors, by using the free composition (see here).
Now ...
15
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1
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686
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Amenability of groups in terms of a perturbation condition
Let $G$ be a countable group and $\lambda \colon G \to U(\ell^2 G)$ its left-regular representation. Suppose that there exists a constant $C>0$ such that for all $T \in B(\ell^2 G)$
$$\inf \lbrace\...
- 25.5k
3
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0
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109
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Two questions on topological and geometric structure of projections in a simple $C^{*}$ algebra
Let $A$ be a simple $C^{*}$ algebra. Assume that the space of projections has a connected component homeomorphic to the complex Grassmanian $G(k,n)$. Is it true to say that, for all $k'<k$, the ...
- 356
1
vote
1
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199
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not commutative symmetric vs strong symmetric spaces
Let M be a von Neumann algebra with semi-finite normal faithful trace $\tau$, $S(M)$ is space of all measurable operators introduced by I.Segal.
For the self-adjoint measurable operator $X\eta M$ ...
- 21
3
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0
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431
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Bohr topos and quantization
Bohrification is a natural way to construct a quantum "phase space" (with some nice insights on foundational problems like non-contextuality through Kochen-Specker etc). I was wondering, since we get ...
- 585
8
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1
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813
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Tomita-Takesaki theory for a simple class of crossed products
This question is inspired by the construction of the time evolution for endomotives as given by Connes and Marcolli in their book http://www.alainconnes.org/docs/bookwebfinal.pdf.
Let $M$ be a monoid ...
- 2,029
5
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1
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289
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Unitary representation acting on the K-theory of the reduced group $C^*$-algebra
Let $G$ be a group (usually infinite), $R$ a ring and $\rho: G \rightarrow Gl_n(\mathbb{Z})$ a finite-dimensional representation of $G$. Then we can define a functor from the category of projective $...
- 976
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1
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427
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Separability of Hilbert spaces from GNS construction.
Suppose we have type $II_1$ factor $\mathcal{M}$ acting on separable Hilbert space $H$. Consider a faithful tracial state $f=tr$ (we know that such object exists) and produce $H_f$ as a Hilbert space
...
3
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0
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90
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Isometric domain of a unital completely positive map with respect to $L_p$-norms
This question can be formulated for general ($\sigma$-finite) von Neumann algebras, but for me it is enough to consider matrix algebras.
So let $M$ be a matrix algebra and $\rho$ a faithful state (...
- 31
2
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0
answers
400
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How to determine there exists a unique invariant subspace for a set of matrices
Hi everyone,
Ive been looking at the following problem, but its not entirely in my area and some potential solutions seem to rely on algebraic geometry. Maybe thats just a complicated way to solve ...
- 39
2
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1
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137
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Continuous depedence of the spectrum on elements
Suppose $a_n \to a$ in a unital C*-algebra $A$. If $\lambda_n \in \sigma(a_n)$ and $\lambda_n \to \lambda$, then $\lambda \in \sigma(a)$. Does the converse hold?
So if $\lambda \in \sigma(a)$, does ...
- 388
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0
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250
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Fusion categories with permutation "associativity matrices"
Let $\mathcal{C}$ be a fusion category and let $(H_1,...,H_r)$ be its simple objects.
$\mathcal{C}$ is non-pointed if at least one of its simple object has Perron-Frobenius dimension $ \neq 1$.
...
1
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1
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67
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Comparision of two $C^{*}$ algebras associated to a non vanishing vector field on a compact manifold
Let $X$ be a non vanishing vector field on a compact manifold $M$ so we have a one dimensional foliation $F$ of $M$ with orbits of $X$.
This foliation defines a $C^{*}$ algebra $C^{*}(F)$. On the ...
- 356
7
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1
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380
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Clarifying the link between deformation/rigidity and dual cocycles
Suppose that a type $II_{1}$ factor $M$ decomposes in two ways as a group von Neumann algebra, e.g. as $L\Gamma$ and as $L\Lambda$. The decomposition $L\Gamma$ gives rise to a comultiplication $$\...
- 7,057
1
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1
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134
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Is there a Frobenius reciprocity for the coproduct?
Let $\mathcal{P}$ be an irreducible finite index-depth subfactor planar algebra. The $2$-boxes space $\mathcal{P}_{2,+}$ is equipped with the coproduct $(a,b) \mapsto a*b = \mathcal{F}(\mathcal{F}^{-...
1
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0
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138
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Noncommutativization of fixed point theory
What papers or references have been devoted for a noncommutativization of "Fixed point theory". Here the terminology Noncommutativiztion, as usual, indicates to that famous table with 2 columns: ...
- 356
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0
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293
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Lifting triangles in K-theory to KL-groups
Let $X$ and $Y$ be finite simplicial complexes (or $CW$-complexes) so that $Y\subseteq X$. Let $s\colon C(X)\to C(Y)$ be the map given by restriction. In particular $K_{*}(C(X))$ and $K_{*}(C(Y))$ are ...
3
votes
2
answers
564
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Positive extension of functionals on a subset of the state space of a $C^*$ algebra
Let $A$ be a finite dimensional $C^*$ algebra and $S(A)$ the state space. Let $K\subset A$ be an intersection of $S(A)$ with a vector subspace $J\subset A$ and let $f$ be a positive affine functional ...
- 41
13
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1
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481
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Can the minimal index of a subfactor take all values in {4cos^2(pi/n);n=3,4,5,...} u [4,infinity]?
Given a subfactor $N\to M$ and a conditional expectation $E:M\to N$,
there is a numerical invariant $Ind(E)$ associated to to this situation, called the index of $E$.
The possible values of $Ind(E)$ ...
- 43.2k
2
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1
answer
391
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when does a $C^*$-algebra have no nonzero unital quotient?
In their paper: "Addition of $C^*$-algebra extensions", G. A. Elliott and D. E. Handelman have discussed some relation between traces and equivalence of projections in $M(A)$, where $M(A)$ is the ...
- 147
0
votes
3
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420
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Existence of tensor product of subalgebras
Let $\mathcal{G} = \mathbb{M}_n(\mathbb{C})$ be an $n$-by-$n$ matrix algebra over complex numbers. Next let $\mathcal{A} \cong \mathbb{M}_d(\mathbb{C})$ be a subalgebra of $\mathcal{G}$ and assume $d$ ...
- 133
2
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1
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286
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Reference request: tensor products of states separate the points of tensor product of $C^*$-alagebras
Suppose $A\otimes B$ is the minimal tensor product of two unital $C^*$ algebras $A$ and $B$.
We know that the set of states, $\{\phi\otimes\psi|\phi\in S(A),\psi\in S(B) \}$ on $A\otimes B$ ...
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